Method of determining an uncertainty estimate of an estimated velocity

ABSTRACT

A method of determining an uncertainty estimate of an estimated velocity of an object includes, determining the uncertainty with respect to a first estimated coefficient and a second estimated coefficient of the velocity profile equation of the object. The first estimated coefficient being assigned to a first spatial dimension of the estimated velocity and the second estimated coefficient being assigned to a second spatial dimension of the estimated velocity. The velocity profile equation represents the estimated velocity in dependence of the first estimated coefficient and the second estimated coefficient. The method also includes determining the uncertainty with respect to an angular velocity of the object, a first coordinate of the object in the second spatial dimension, and a second coordinate of the object in the first spatial dimension.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit under 35 U.S.C. § 119(a) of EuropeanPatent Application EP 18189410.6, filed Aug. 16, 2018, the entiredisclosure of which is hereby incorporated herein by reference.

TECHNICAL FIELD OF INVENTION

The invention relates to a method of determining an uncertainty estimateof an estimated velocity of an object, wherein the uncertainty estimaterepresents an uncertainty of the estimated velocity with respect to atrue velocity of the object.

BACKGROUND OF INVENTION

Uncertainty estimates are useful for quantifying the validity of anestimated velocity. In other words, an uncertainty measure representsthe extent to which an estimated velocity can be trusted, or theprobability that the estimated velocity matches the true velocity. Thisis, the uncertainty can be interpreted as the range of potential errorbetween the true and the estimated velocity. According to one definitionthe uncertainty represents a range of velocities wherein the truevelocity is assumed to fall into the range. Accordingly, the uncertaintyestimate is an estimate of this range.

SUMMARY OF THE INVENTION

In general, the higher the potential error between the estimated and thetrue velocity, the higher the uncertainty. Preferably, the uncertaintyestimate is at least correlated to the uncertainty, which may manifestin that the uncertainty is proportional to the uncertainty estimate.

In modern automotive applications, in particular automated or autonomousdriving applications, there is the need to exactly determine motioncharacteristics of any objects in the vicinity of a host vehicle. Suchobjects can be other vehicles but also pedestrians or stationary objectslike traffic signs or walls. With the position and the motioncharacteristics of any objects in the vicinity of the host vehicle, itis possible to determine the environment of the vehicle in the sense ofa general perception by means of sensor technology operably installed inthe host vehicle. It is understood that the host vehicle (i.e., acontrol unit of the host vehicle) needs to have precise informationabout the velocities of surrounding objects in order to be able tosecurely control the vehicle by automatically generated drivingcommands. However, as indicated above, a totally precise estimation ofthe velocities of surrounding objects is usually not possible. In orderto still allow secure automatic driving of the host vehicle, uncertaintyestimates for each estimated velocity are of great help since thecontrol unit of the host vehicle can readily adapt itself to thevalidity of the estimated velocities, thereby enabling optimum use ofthe available technical information.

An important motion characteristic is the velocity of a given object,which usually needs to be estimated by the host vehicle by means ofsensors and is therefore subject to an estimation error, i.e. theestimated velocity deviates from the true velocity of the object. Oneway to determine an estimated velocity is by using one or more radarsensors installed in the host vehicle, wherein a plurality of detectionpoints are determined with a radar system, each detection pointrepresenting an estimated velocity for a given location in the vicinityof the host vehicle. A detection point can also be located on an objectand can thus serve for estimating the velocity of the object.Preferably, a plurality of detection points on a single object arejointly evaluated in order to derive an estimated velocity which is moreprecise than the estimated velocity of a single detection point. Anestimated velocity can comprise magnitude and direction, so that theestimated velocity is a vector with two components, i.e. quantifying thevelocity with respect to two spatial dimensions. However, it is alsopossible that the estimated velocity only comprises a magnitude value.

Due to the inherent limitations of obtaining precise estimatedvelocities with modern sensor technology, in particular radartechnology, there is the need to obtain knowledge about the potentialerror of an estimated velocity. In this regard, an uncertainty measurecan be used for further processing of an estimated velocity, for examplefor a tracking algorithm configured to track the object so that the hostvehicle has precise information about the motion of the particularobject. This is to say that a given value of an estimated velocity isprocessed together with an uncertainty estimate. In this way, it can beensured that the estimated velocity influences a given application independence of its validity. In more simple terms, an estimated velocitywith a high uncertainty estimate should have less influence than anotherestimated velocity with a low uncertainty estimate.

There is the need for accurate uncertainty estimates (or measures) whichquantify the true uncertainty of estimated velocities as best aspossible, in particular when the object is rotating with an angularvelocity that cannot or should not be estimated. In some applications,the variance of an estimated velocity can be used as an uncertaintyestimate. However, it has been found that such an uncertainty estimateis not accurate in many motion situations which appear in real trafficscenarios. In particular, if a given object is not moving along astraight line (linear motion), variance is often a very bad uncertaintyestimate. Furthermore, variance often shows a systematic error whichbadly influences any further processing of an estimated velocity, thusleading to a reduced performance of automated driving applications. Thisis not tolerable as the security for any traffic participants cannot besubject to a compromise.

A problem to be solved by the invention is to provide an improved methodof determining an uncertainty estimate for an estimated velocity of anobject. The problem is solved by the method according to claim 1.

A method of determining an uncertainty estimate of an estimated velocityof an object comprises the following steps: (a) determining a firstportion of the uncertainty estimate, the first portion representing theuncertainty with respect to a first estimated coefficient and a secondestimated coefficient of the velocity profile equation of the object,the first estimated coefficient being assigned to a first spatialdimension of the estimated velocity and the second estimated coefficientbeing assigned to a second spatial dimension of the estimated velocity.The velocity profile equation represents the estimated velocity independence of the first estimated coefficient and the second estimatedcoefficient;

(b) determining a second portion of the uncertainty estimate, the secondportion representing the uncertainty with respect to an angular velocityof the object, a first coordinate of the object in the second spatialdimension and a second coordinate of the object in the first spatialdimension; and

(c) determining the uncertainty estimate on the basis of the firstportion and the second portion.

In summary, the uncertainty measure comprises or is composed of twodedicated portions. The first portion represents the uncertainty withrespect to the two estimated “velocity” coefficients of the so calledvelocity profile equation, wherein the velocity profile equation isgenerally known from the art (cf. D. Kellner, M. Barjenbruch, K.Dietmayer, J. Klappstein, J. Dickmann, “Instantaneous lateral velocityestimation of a vehicle using Doppler radar,” Information Fusion(FUSION), 2013 16th International Conference, Istanbul, 2013). Thisequation is also known as range rate equation and generally representsthe estimated velocity in dependence of the two said coefficients. Thesecoefficients are assigned to respective spatial dimensions, which meansthat the coefficients generally represent velocity components of anobject in these dimensions. This interpretation is usually bound to amotion scenario in which the object is moving along a straight line(linear movement). However, if there is a rotational movement about apredefined axis of the object, i.e. the object has a yaw rate greaterthan zero, the coefficients do not fully represent “linear” velocitycomponents but a mixture of rotational and linear velocity portions.Nevertheless, the coefficients are assigned to said dimensions becausethey are preferably used to evaluate the total velocity component ineach dimension. The velocity profile equation is generally well known tothe skilled person in the field of estimating motion of objects by meansof sensor technology, in particular radar technology. The velocityprofile equation is also discussed and explicitly provided furtherbelow.

The first and second estimated coefficient of the velocity profileequation are estimated figures and contribute to the overall uncertaintyof the estimated velocity. Said first portion of the uncertainty measurecan thus be interpreted to capture the uncertainty information that isconnected to the coefficients.

The second portion of the uncertainty estimate is connected to theangular velocity of the object. In particular, the angular velocity is ayaw rate of the object, i.e., a rotational velocity of the object in ahorizontal plane about a predefined axis which extends orthogonally tosaid plane. For example, the yaw rate of a car (one type of object)generally corresponds to a lateral movement of the car caused bysteering activities of the driver while travelling.

An estimation of the angular velocity usually delivers insufficientresults when using only one scan (i.e. one measurement instance) of onesensor. This is to say that the estimated angular velocity is in mostcases not plausible. The estimation can be improved by filtering signalsover time or by using two sensors (cf. D. Kellner, M. Barjenbruch, J.Klappstein, J. Dickmann, and K. Dietmayer, “Instantaneous full-motionestimation of arbitrary objects using dual doppler radar,” inProceedings of Intelligent Vehicles Symposium (IV), Dearborn, Mich.,USA, 2014). However, this solution requires multiple sensors forcovering the field of view and leads to a significant increase of theoverall costs.

The angular velocity of the object contributes to the overalluncertainty of the estimated velocity of the object (independent from apotential estimation of the angular velocity). It can be shown that ingeneral terms a mathematical connection exists between the estimatedvelocity, the first and second coefficients, and the angular velocity.This connection can be expressed by a first coordinate of the object inthe second spatial dimension and a second coordinate of the object inthe first spatial dimension. The first and second coordinates arerelated to or represent a point which corresponds to a location on theobject for which the estimated velocity is assumed to be valid. However,the coordinates are inverted between the dimensions, which is due tomathematical reasons, as will become apparent in further detail below.The proposed uncertainty estimate takes these coordinates into account,which leads to a more accurate uncertainty estimate in view this localdefinition of the second portion.

The final uncertainty estimate is formed on the basis of the first andsecond portion of the uncertainty estimate. The split-up of theuncertainty estimate into two portions has been found to lead to ahigher accuracy of the overall estimate. In particular, treating theangular velocity separately from the coefficients of the range rateequation is one aspect that improves the reliability of the proposeduncertainty estimate.

Advantageous embodiments of the invention are specified in the dependentclaims, the description and the drawings.

According to one embodiment the uncertainty estimate represents adispersion of the estimated velocity. Likewise, the first portion of theuncertainty estimate represents a dispersion of the first estimatedcoefficient and the second estimated coefficient. Furthermore, thesecond portion can represent a dispersion of the angular velocity of theobject. The term “dispersion” is understood as a spread in the sensethat it indicates a range of possible values. Known types of dispersionare variance and standard deviation, also the term dispersion is notrestricted accordingly. These types represent a dispersion of valuesaround a mean. The advantage of expressing the uncertainty measure orparts of it in terms of a dispersion is an intuitive understanding ofthe estimate, which is known in the field of statistics. For example ahigh dispersion may represent a high uncertainty.

According to another embodiment the uncertainty estimate and/or thefirst portion and/or the second portion of the uncertainty estimate aredetermined as a two-dimensional matrix, wherein the two-dimensionalmatrix represents a dispersion with respect to the first spatialdimension and the second spatial dimension. In particular, each of theportions and the resulting uncertainty estimate can be determined, i.e.expressed as a two-dimensional matrix. The use of matrices isadvantageous in view of an efficient processing of a given estimatedvelocity for which the uncertainty estimate should be valid.Furthermore, a direct correspondence between the two spatial dimensionsand the two dimensions of the matrix can be implemented.

According to another embodiment the first portion is determined on thebasis of:

-   -   a covariance portion representing a covariance matrix of the        first estimated coefficient and the second estimated        coefficient; and    -   a bias portion representing a bias of the first estimated        coefficient and/or the second estimated coefficient.

For example the first portion of the uncertainty measure can bedetermined as the sum of the covariance portion and the bias portion.However, other types of combination are also possible.

Said covariance portion can be determined as the covariance matrix ofthe first and second estimated coefficients, wherein the covariancematrix includes the variances of the coefficients and the covariancesbetween them. This structure of a covariance matrix is known from thefield of statistics and it can be calculated with high efficiency. Ithas proven robust and it enhances the accuracy of the uncertaintyestimate in combination with other portions.

The term “bias” is generally interpreted as a systematical statisticalerror. For example, the bias can be a constant representing an averagedifference between an estimated mean and a true mean of values. Thecovariance portion can be centered around an estimated mean, wherein thebias can indicate how far the estimated mean is away from the true mean.Therefore, the bias portion can be interpreted as a correction portionwith respect to a systematical error in the covariance portion oranother (sub-) portion of the first portion.

It is important to point out that the bias (or estimated bias) is notused to correct the estimated velocity but rather to improve theuncertainty estimation so as to make it more consistent. The bias isestimated so as to improve the validity of the estimated uncertainty(not the estimated velocity). A precise knowledge of the bias is notnecessary and in some cases cannot even be estimated due to a lack ofobservability. This lack can be handled by introducing the bias portion.A similar approach could also be used to determine the uncertainty ofthe angular velocity.

Having further regard to the bias portion, according to one embodiment,the bias portion can be determined on the basis of a plurality ofdetection points of the object and at least one constant, wherein eachof the plurality of detection points comprises an estimated velocity ata detection position on the object, said detection position on theobject being defined at least by an angle. The detection points can beacquired by means of a radar system as pointed out further above,however, other sensor technology is also possible. The detection pointsare preferably acquired in one time instance, i.e. the plurality ofdetection points only comprise detection points from one scan, inparticular a radar scan of a radar system.

The plurality of detection points allows to adapt the bias portion independence of actual sensor data, which significantly improves theaccuracy of the uncertainty estimate. In particular, the bias portioncan use the same data as used for determining the first and secondcoefficients of the velocity profile equation, i.e. the data from thedetection points is used for determining the coefficients of thevelocity profile equation an the bias portion.

The estimated velocity (for which the uncertainty estimate isdetermined) can be assigned to a position of the object. This positionis preferably represented by the first coordinate of the object in thefirst spatial dimension and the second coordinate of the object in thesecond spatial dimension and wherein the uncertainty estimate isdetermined in dependence of said position of the object. Assigning theuncertainty to a particular position on or of the object improves theaccuracy of the uncertainty measure because its “maximum local validity”is explicitly taken into account so that subsequent processing of theestimated velocity can benefit from this information. “Lossy averaging”,which could be done in order to provide one single uncertainty estimatefor a large object can be avoided.

The estimated velocity (for which the uncertainty estimate isdetermined) can be equal to the velocity of the velocity profileequation, i.e. the estimated velocity can directly depend on the firstand second estimated coefficient. However, the estimated velocity canalso be determined in other way, for example by using other sensortechnology.

In the following, embodiments with regard to determining the secondportion will be addressed.

According to one embodiment the second portion is determined on thebasis of an intermediate second portion. The intermediate second portionrepresents the uncertainty with respect to the angular velocity only,wherein the intermediate second portion is predetermined. In otherwords, the intermediate second portion only represents the “angularvelocity uncertainty”, e.g. a range of values which is assumed toinclude or cover the true unknown angular velocity. In this approach,the intermediate second portion is predetermined, i.e. is set to one ormore predefined values. It has been found that better results can beachieved with a predetermined intermediate second portion, rather thantrying to determine the intermediate second portion on the basis of realestimates of the angular velocity. Angular velocity is difficult toestimate with high precision, in particular when the angular velocityshould be estimated from only one scan (“single processing instance”).The estimation also remains to be difficult when multiple sensors areused to observe one single object (cf. D. Kellner, M. Barjenbruch, J.Klappstein, J. Dickmann, and K. Dietmayer, “Instantaneous full-motionestimation of arbitrary objects using dual doppler radar,” inProceedings of Intelligent Vehicles Symposium (IV), Dearborn, Mich.,USA, 2014). When only one sensor is used for observing an objectestimation of the angular velocity (i.e. yaw rate) of the object isextremely difficult, in most cases the estimated angular velocity cannoteven be treated as roughly plausible. In this context, the proposeduncertainty measure can provide significant advantages because anestimation of the angular velocity can be completely avoided.

Relying on a predetermined intermediate second portion can even lead tomuch more accurate results because it has been recognized that angularvelocity is usually bounded to certain limits for most object classes.For example in a traffic scenario, it can be assumed that the angularvelocity of any object will usually be below a certain threshold.

In a preferred embodiment, the intermediate second portion ispredetermined by the variance of a distribution of the angular velocityof the object. This is an approach for modelling the angular velocity bymeans of an assumed distribution. The variance or related figures can bepicked as a key figure because it is well suited to represent a range ordispersion of values, which is line with the general mindset of theuncertainty estimate. It is possible that the intermediate secondportion is equal to a predetermined variance of the angular velocity ofthe object.

In a specific embodiment, said distribution is a uniform distributionwith at least one predetermined extremum of the angular velocity of theobject. Therefore, it is possible to make an explicit model assumptionvia a distribution. Other model distributions for the angular velocityare also possible, for example triangular or trapezoidal distributions.

In another embodiment, the intermediate second portion is predeterminedby at least one parameter representing an extremum of the angularvelocity of the object. For example, a maximum value for the angularvelocity can be manually set and used for parameterizing theintermediate second portion. This can be done, e.g., by determining thesecond intermediate portion as the variance of a model distribution ofthe angular velocity, wherein the model distribution is limited toangular velocities between the negative maximum angular velocity(negative extremum) and the positive maximum angular velocity (positiveextremum). As pointed out before, a uniform distribution may be chosenas a model distribution, but other model distributions are alsopossible. This may depend on the particular application. It is alsopossible to choose the model distribution in dependence of an objecttype. For example, if the object is automatically classified as apassenger car a different distribution may be chosen than if the objectis classified as a lorry. Likewise, different extrema may be chosen forthe intermediate second portions. So in general terms, determination ofthe uncertainty estimate may depend on a class of the object, whereinthe class can be determined automatically by using a classificationmethod. Such a classification method may be based on visual data of theobject acquired by means of a camera but may also be based on theestimated velocity of the object.

In yet another embodiment, the uncertainty estimate can be determined onthe basis of a sum of the first portion and the second portion. Othermathematical combinations are also possible, for example a quotientbetween the first portion and the second portion of the uncertaintyestimate.

Having regard to possible uses of the uncertainty estimate the methodcan further comprise controlling a vehicle in the vicinity of the objectin dependence of the estimated velocity of the object, wherein theestimated velocity is processed in dependence of the uncertaintyestimate.

The invention also relates to a storage device with software, inparticular firmware, for carrying out method the method according to oneof the preceding embodiments.

The storage device can be part of a system comprising the storage deviceand a vehicle, wherein the vehicle comprises a sensor and a control unitconfigured to cooperate with the storage device. The control unit isfurther configured to determine a plurality of detection points of atleast one object in the vicinity of the sensor, each detection pointcomprising an estimated velocity at a position on or near the object.Moreover, the control unit is configured to determine an uncertaintyestimate of the estimated velocity for at least one of the detectionpoints by using the software stored in the storage device.

In an embodiment of the system the control unit can be furtherconfigured to track the object on the basis of the uncertainty estimateand/or to classify the object on the basis of the uncertainty estimate.It is also possible to discriminate the object from other objects on thebasis of the uncertainty estimate. Other applications, in particularautomotive application, in which estimated velocities of objects otherthan the (host) vehicle are employed, can also be modified such that theuncertainty estimate is taken into account. The reliability and accuracyof such applications can thus be improved. Examples of such applicationsare distance control, valet parking and autonomous driving.

According to another embodiment of the system the sensor and the controlunit can form a so called Pulse-Doppler radar system, which is awidespread and well-known system for determining a plurality ofdetection points, each detection point representing an estimatedvelocity at a position on the object, and the position being defined atleast by an angle θ_(i). The angle is often an azimuth angle as itrepresents an angle about a boresight of a radar antenna of the system,wherein the angle is in a horizontal plane, which corresponds to aground beneath the vehicle.

As a general aspect of the disclosure and according to an embodiment ofthe method the first estimated coefficient and the second estimatedcoefficient are determined on the basis of a plurality of detectionpoints, each detection point representing an estimated velocity at aposition on the object, the position being defined at least by an angleθ_(i), wherein the velocity profile equation is represented by:

${{\overset{.}{r}}_{i,{cmp}} = {\begin{bmatrix}{\cos\;\theta_{i}} & {\sin\;\theta_{i}}\end{bmatrix}\begin{bmatrix}{\overset{\sim}{c}}_{t} \\{\overset{\sim}{s}}_{t}\end{bmatrix}}},$wherein {dot over (r)}_(i,cmp) represents the velocity of the object atthe position of the i-th detection point, {tilde over (c)}_(t) denotesthe first estimated coefficient, {tilde over (s)}_(t) denotes the secondestimated coefficient, and θ_(i) denotes the position of the i-thdetection point. The plurality of detection points are preferablyacquired in one time instance, i.e. the plurality of detection pointsonly comprise detection points from one scan, in particular a radar scanof a radar system.

A radar system, in particular said Pulse-Doppler radar system is wellsuited to provide a plurality of detection points on which basis saidvelocity profile equation can readily be determined. The proposeduncertainty estimate is particularly suitable to accurately representthe uncertainty of estimated velocities determined by using the velocityprofile equation.

It is understood that in connection with mathematical expressionsdisclosed herein that mathematical expressions do not necessarily needto be exactly fulfilled in a strict mathematical sense. In particular,an algebraic expression can be understood in a conceptual sense. This isto say that an equal sign can still be satisfied if the equality is onlyapproximately fulfilled. Therefore, if the expression is implemented ona computer machine any numerical deviations from the narrow meaning ofthe expression (i.e., offsets or essentially constant factors) which aremerely due to technical details of the implementation do not influencethe fact that the implementation falls under the meaning of theexpression, as is understood by those skilled in the art. In particular,any equality sign (i.e., “=”) that appear in any of the algebraicexpressions disclosed herein may be substituted by a proportionalitysign (i.e., “˜”).

BRIEF DESCRIPTION OF DRAWINGS

The invention is further described by way of example with reference tothe drawings in which:

FIG. 1 shows a target coordinate system;

FIG. 2 shows a vehicle coordinate system;

FIG. 3 shows a sensor coordinate system;

FIG. 4 shows a target vehicle with respect to a host vehicle withdetection points located on the target vehicle;

FIG. 5 illustrates how to calculate velocity vectors at the location ofa detection point; and

FIG. 6 illustrates an embodiment of the method as described herein.

DETAILED DESCRIPTION

Generally, a host vehicle 4 (see FIG. 2) is equipped with a radar system5′ (see FIG. 2) where reflected radar signals from a target 2 (FIG. 1)in the field of view of the radar system 5′ are processed to providedata in order to ascertain parameters used in the methodology.

In order to do this various conditions and requirements may be ofadvantage. The target 2 (rigid body, e.g. vehicle) is preferably anextended target, i.e., the target allows the determination of aplurality of points of reflection 6′ (see FIG. 4) that are reflectedfrom the target 2 in real-time and that are based on raw radar detectionmeasurements.

The target 2 is an example of an object in the sense of the general partof the description and the claims. However, other types of objects arealso possible, in particular objects that appear in ordinary trafficscenes, like motorcycles, bicycles, pedestrians, large as well as smallvehicles. Moreover, in principal objects can also be stationary objects.

As used herein, the term “extended target” is used to refer to targets 2that are capable of providing multiple, i.e. two, three or morespaced-apart scattering-points 6′ also known as points of reflection 6′.The term “extended target” is thus understood as a target 2 that hassome physical size. In this instance, it should be noted that thephysical size can be selected, e.g., in the range of 0.3 m to 20 m inorder to be able to detect points of reflection 6′ stemming from, e.g.,a moving person to a moving heavy goods vehicle or the like.

The various scattering points 6′ are not necessarily individuallytracked from one radar scan to the next and the number of scatteringpoints 6′ can be a different between scans. Furthermore, the locationsof the scattering points 6′ can be different on the extended target 2 insuccessive radar scans.

Radar points of reflection 6′ can be determined by the host vehicle 4from radar signals reflected from the target 2, wherein a comparison ofa given reflected signal with an associated emitted radar signal can becarried out to determine the position of the radar point of reflection6′, e.g., in Cartesian or Polar coordinates (azimuth angle, radialrange) with respect to the position of a radar-emitting and/orradar-receiving element/unit on the host vehicle, which can be theposition of the radar sensor unit.

By using, e.g., Doppler radar techniques, the range rate is alsodetermined as known in the art. It is to be noted that the “raw data”from a single radar scan can provide the parameters θ_(i) (azimuthangle) and {dot over (r)}_(i) (raw range rate, i.e., radial velocity)for the i-th point of reflection of n points of reflection. These arethe parameters which are used to estimate the velocity of a (moving)target, wherein i=1, . . . , n.

It is also to be noted that the term instantaneous radar scan, singleradar scan or single measurement instance can include reflection datafrom a “chirp” in Doppler techniques, which may scan over, e.g., up to 2ms. This is well known in the art. In the subsequent description, thefollowing conventions and definitions are used:

World Coordinate System

As a convention, a world coordinate system with the origin fixed to apoint in space is used it is assumed that this world coordinate systemdoes not move and does not rotate. Conventionally, the coordinate systemis right-handed; the Y-axis, orthogonal to the X-axis, pointing to theright; the Z-axis pointing into the page and a an azimuth angle isdefined in negative direction (clock-wise) with respect to the X-axis;see FIG. 1 which shows such a coordinate system with origin 1 and anon-ego vehicle target 2. FIG. 1 further shows the extended target 2 inthe form of a vehicle, e.g. an object having a length of approximately4.5 m.

Vehicle Coordinate System

FIG. 2 shows a vehicle coordinate system that in the present instancehas its origin 3″ located at the center of the front bumper 3 of a hostvehicle 4. It should be noted in this connection that the origin 3″ ofthe vehicle coordinate system can be arranged at different positions atthe host vehicle 4.

In the present instance the X-axis is parallel to the longitudinal axisof the host vehicle 4, i.e. it extends between the front bumper 3 and arear bumper 3′ and intersects with the center of the front bumper 3 ifthe origin 3″ is located there. The vehicle coordinate system isright-handed with the Y-axis orthogonal to the X-axis and pointing tothe right, the Z-axis pointing into the page. An (azimuth) angle isdefined as in the world coordinate system.

Sensor Coordinate System

FIG. 3 shows a sensor coordinate system having the origin 5. In theexample of FIG. 3 the origin 5 is located at the center of a radarsystem (in particular sensor unit) 5′, which can be a radome. The X-axisis perpendicular to the sensor radome, pointing away from the radome.The coordinate system is right-handed: Y-axis orthogonal to the X-axisand pointing to the right; Z-axis pointing into the page. An (azimuth)angle is defined as in the world coordinate system.

The velocity and the yaw rate of the host vehicle 4 are assumed to beknown from sensor measurements known in the art. The over-the-ground(OTG) velocity vector of the host vehicle 4 is defined as:V _(h)=[u _(h) v _(h)]^(T),where u_(h) is the longitudinal velocity of the host vehicle 4 (i.e.,the velocity in a direction parallel to the X-axis of the vehiclecoordinate system) and v_(h) is lateral velocity of the host vehicle 4(i.e., the velocity in a direction parallel to the Y-axis of the vehiclecoordinate system). In more general terms, the longitudinal velocity andthe lateral velocity are a first and a second velocity component of thehost vehicle 4, respectively. The X-axis and the Y-axis generallycorrespond to a first spatial dimension and a second spatial dimension,respectively. Likewise, the longitudinal direction and the lateraldirection correspond to the first spatial dimension and the secondspatial dimension, respectively. These are preferably but notnecessarily in orthogonal relation to each other.

The sensor mounting position and boresight angle with respect to thevehicle coordinate system are assumed to be known with respect to thevehicle coordinate system (VCS), wherein the following notations areused:

-   -   x_(s,VCS)—sensor mounting position with respect to longitudinal        (X-) coordinate    -   y_(s,VCS)—sensor mounting position with respect to lateral (Y)        coordinate    -   y_(s,VCS)—sensor boresight angle.

The sensor over-the-ground (OTG) velocities can be determined from theknown host vehicle velocity and the known sensor mounting position. Itis understood that more than one sensor can be integrated into onevehicle and specified accordingly.

The sensor OTG velocity vector is defined as:V _(s)=[u _(s) v _(s)]^(T),wherein u_(s) is the sensor longitudinal velocity and v_(s) is thesensor lateral velocity corresponding generally to first and secondvelocity components in the case of a yaw rate of zero.

At each radar measurement instance (scan) the radar sensor unit capturesn (raw) detection points from the target. Each detection point i=1, . .. , n can be described by the following parameters expressed in thesensor coordinate system:

-   -   r_(i)—range (or radial distance),    -   θ_(i)—azimuth angle,    -   {dot over (r)}_(i)—raw range rate (or radial velocity).

Target planar motion can be described by the target OTG velocity vectorat the location of each raw detection:V _(t,i)=[u _(t,i) v _(t,i)]^(T),wherein u_(t,i) represents the longitudinal velocity of the target atthe location of the i-th detection point and v_(t,i) represents thelateral velocity of the target at the location of the i-th detectionpoint, both preferably but not necessarily with respect to the sensorcoordinate system.

Target planar motion can be described as well by:V _(t,COR)=[ω_(t) x _(t,COR) y _(t,COR)]^(T),wherein ω_(t) represents the yaw rate (angular velocity) of the target,x_(t,COR) the longitudinal coordinate of the center of target's rotationand y_(t,COR) the lateral coordinate of the center of target's rotation.

FIG. 4 illustrates target velocity vectors as lines originating from aplurality of detection points 6′ illustrated as crosses, wherein thedetection points 6′ are all located on the same rigid body target 2 andwherein the detection points 6′ are acquired using a sensor unit of ahost vehicle 4.

The general situation is shown in greater detail in FIG. 5 showing threedetection points 6 located on a target (not shown) with a center ofrotation 7. The vehicle coordinate system with axes X_(VCS), Y_(VCS) isshown in overlay with the sensor coordinate system having axes X_(SCS),Y_(SCS). The velocity vector of one of the detection points 6 (i=1) isshown together with its components u_(t,i), v_(t,i).

The range rate equation for a single detection point 6 can be expressedas follows:{dot over (r)} _(i) +u _(s) cos θ_(i) +v _(s) sin θ_(i) =u _(t,i) cosθ_(i) +v _(t,i) sin θ_(i),wherein {dot over (r)}_(i) represents the range rate, i.e., the rate ofchange of the distance between the origin of the sensor coordinatesystem and a detection point 6, as illustrated in FIG. 5. The locationof the detection point 6 can be described by the azimuth angle θ_(i=1)and the value of the radial distance r_(i=1) (range of detection point,i.e. distance between origin and the detection point).

To simplify the notation the compensated range rate can be defined as:{dot over (r)} _(i,cmp) ={dot over (r)} _(i) +u _(s) cos θ_(i) +v _(s)sin θ_(i)with {dot over (r)}_(i,cmp) representing the range rate of the i-thdetection point compensated for the velocity of the host vehicle 4.

The compensated range rate can also be expressed as:{dot over (r)} _(i,cmp) =u _(t,i) cos θ_(i) +v _(t,i) sin θ_(i).

The compensated range rate can also be expressed in vector notation as:

${\overset{.}{r}}_{i,{cmp}} = {{\begin{bmatrix}{\cos\;\theta_{i}} & {\sin\;\theta_{i}}\end{bmatrix}\begin{bmatrix}u_{t,i} \\v_{t,i}\end{bmatrix}}.}$

The so called velocity profile equation (or range rate equation) isdefined as:{dot over (r)} _(i,cmp) =c _(t) cos θ_(i) +s _(t) sin θ_(i),wherein c_(t) represents the first, e.g. longitudinal, coefficient orcomponent of the range rate and s_(t) represents the second, e.g.lateral, coefficient or component of the range rate equation. Note thatthe coefficients c_(t), s_(t) are preferably invariant with respect tothe azimuth angle at least for a range of azimuth angles correspondingto the location of the target to which a plurality of detection pointsrefer to and on which basis the coefficients have been determined. Thismeans that the velocity profile equation is assumed to be valid not onlyfor specific detection points but for a range of azimuth angles.Therefore, the range rate can readily be determined for any azimuthangle from a specific angle range using the range rate equation. Therange rate is an example of an estimated velocity in the general senseof the disclosure.

As the skilled person understands, in practice, the “true” coefficientsc_(t), s_(t) are usually estimated from a plurality of detection points.These estimates are denoted c_(t) and {tilde over (s)}_(t) and areestimated using, e.g., an iteratively (re-) weighted least squaresmethodology. In the following, an example method for estimating thecoefficients c_(t), s_(t) is described.

Step 1: In an initial step, the method comprises emitting a radar signaland determining, from a plurality of radar detection measurementscaptured by said radar sensor unit, a plurality of radar detectionpoints at one measurement instance. Each radar detection point comprisesat least an azimuth angle θ_(i) and a range rate {dot over (r)}_(i),wherein the range rate {dot over (r)}_(i) represents the rate of changeof the distance between the sensor unit and the target at the locationof the i-the detection point (cf. FIG. 4). It is understood that theazimuth angle θ_(i) describes the angular position of the i-th detectionpoint. It is assumed that the plurality of detection points are locatedon a single target (such target is usually referred to as a distributedtarget) as shown in FIG. 4. The target is an object.

Step 2: The compensated range rate {dot over (r)}_(i,cmp) is determinedas:{dot over (r)} _(i,cmp) ={dot over (r)} _(i) +u _(s) cos θ_(i) +v _(s)sin θ_(i),wherein u_(s) represents the first (e.g. longitudinal) velocitycomponent of the sensor unit and wherein v_(s) represents the second(e.g. lateral) velocity component of the sensor unit. The compensatedrange rate is the range rate compensated for the velocity of the hostvehicle. Therefore, the compensated range rate can be interpreted as theeffective velocity of the target at the location of the i-th detectionpoint. The compensated range rate corresponds to an estimated velocityof the target.

Step 3: From the results of steps 1 and 2, an estimation {tilde over(c)}_(t) of the first coefficient c_(t) of the velocity profile equationof the target and an estimation {tilde over (s)}_(t) of the secondcoefficient s_(t) of the velocity profile equation of the target arepreferably determined by using an iteratively reweighted least squares(IRLS) methodology (ordinary least squares would also be possible)comprising at least one iteration and applying weights w_(i) to theradar detection points, wherein the velocity profile equation of thetarget is represented by:{dot over (r)} _(i,cmp) =c _(t) cos θ_(i) +s _(t) sin θ_(i).The IRLS methodology is initialized, e.g., by the ordinary least squares(OLS) solution. This is done by first computing:

${\begin{bmatrix}{\overset{\sim}{c}}_{t} \\{\overset{\sim}{s}}_{t}\end{bmatrix} = {\lbrack {X^{T}X} \rbrack^{- 1}X^{T}{\overset{.}{r}}_{cmp}}},$wherein {dot over (r)}_(cmp) represents the vector of compensated rangerates {dot over (r)}_(i,cmp) for i=1, 2 . . . n. Using{dot over ({circumflex over (r)})} _(i,cmp) ={tilde over (c)} _(t) cosθ_(i) +{tilde over (s)} _(t) sin θ_(i)an initial solution for {dot over ({circumflex over (r)})}_(i,cmp) iscomputed. Then, the initial residual ise _({dot over (r)},i) ={dot over (r)} _(i,cmp) −{dot over ({circumflexover (r)})} _(i,cmp)is computed.

The variance of the residual is then computed as:

${\hat{\sigma}}_{\overset{.}{r}}^{2} = {\frac{\sum_{i = 1}^{n}( e_{\overset{.}{r},i} )^{2}}{n - 2}.}$

Next, an estimation of the variance of the estimations {tilde over(c)}_(t) and {tilde over (s)}_(t) is computed:{circumflex over (σ)}_(VP) ²={circumflex over (σ)}_({dot over (r)}) ²(X^(T) X)⁻¹.wherein

$X = {\begin{bmatrix}{\cos\;\theta_{1}} & {\sin\;\theta_{1}} \\\vdots & \vdots \\{\cos\;\theta_{n}} & {\sin\;\theta_{n}}\end{bmatrix}.}$

With the initial solution, weights w_(i)∈[0; 1] can be computed independence of the residuals, wherein predefined thresholds may be usedto ensure that the weights are well defined.

The weights w_(i) are then arranged in a diagonal matrix W and theestimation of the coefficients of the first iteration is given as:

$\begin{bmatrix}{\overset{\sim}{c}}_{t} \\{\overset{\sim}{s}}_{t}\end{bmatrix} = {\lbrack {X^{T}{WX}} \rbrack^{- 1}X^{T}W{{\overset{.}{r}}_{cmp}.}}$

Step 4: From the solution of the first iteration an estimation {dot over({circumflex over (r)})}_(i,cmp) of the velocity profile is determinedrepresented by:{dot over ({circumflex over (r)})} _(i,cmp) ={tilde over (c)} _(t) cosθ_(i) +{tilde over (s)} _(t) sin θ_(i),wherein the azimuth angle θ_(i) is determined from step 1 and theestimation of the first and second coefficients ĉ_(t) and ŝ_(t) isdetermined from step 3 (initial solution). A new residual is computedas:e _({dot over (r)},i) ={dot over (r)} _(i,cmp) −{dot over ({circumflexover (r)})} _(i,cmp).

The variance of the new residual is then computed as:

${\hat{\sigma}}_{\overset{.}{r}}^{2} = \frac{\sum\limits_{i = 1}^{n}( {\psi( e_{\overset{.}{r},i} )} )^{2}}{( {\sum\limits_{i = 1}^{n}{\psi( e_{\overset{.}{r},i} )}^{\prime}} )^{2}( {n - 2} )}$withψ(e _({dot over (r)},i))=w _(i) e _({dot over (r)},i),wherein ψ(e_({dot over (r)},i))′ represents the first derivative ofψ(e_({dot over (r)},i)) with respect to the residuale_({dot over (r)},i), and wherein n represent the number of detectionpoints.

Next, an estimation of the variance of the estimations {tilde over(c)}_(t) and {tilde over (s)}_(t) is computed as:{circumflex over (σ)}_(VP) ²={circumflex over (σ)}_({dot over (r)}) ²(X^(T) X)⁻¹,wherein the variance may be compared to a stop criterion (e.g., athreshold) in order to decide whether or not a further iteration iscarried out for determining the estimated coefficients {tilde over(c)}_(t) and {tilde over (s)}_(t). In this way, a final solution for thecoefficients {tilde over (c)}_(t) and {tilde over (s)}_(t) can beobtained.

It can be shown that if the target 2 moves along a straight line (linearmovement), the first and second estimated coefficients correspond to aportion of the velocity in the first and second spatial dimensions(i.e., x-direction and y-direction), respectively, this is:V _(t,i) ^(x) ={tilde over (c)} _(t)V _(t,i) ^(y) ={tilde over (s)} _(t),wherein V_(t,i) ^(x) is the velocity component in the x-direction forthe i-th detection point and V_(t,i) ^(y) is the velocity component inthe y-direction for the i-th detection point. In FIG. 5, these velocitycomponents are indicated for one of the detection points 6, namely i=1,wherein V_(t,i) ^(x)=u_(t,i) and V_(t,i) ^(y)=v_(t,i).

In case the target has a non-zero yaw rate, i.e. ω_(t) is not zero, thevelocity components with respect to the first and second spatialdimensions can be expressed as:

${\begin{bmatrix}V_{t,i}^{x} \\V_{t,i}^{y}\end{bmatrix} = \begin{bmatrix}{( {y_{t,{COR}} - y_{t,i}} )\omega_{t}} \\{( {x_{t,i} - x_{t,{COR}}} )\omega_{t}}\end{bmatrix}},$wherein x_(t,i) is a first coordinate of the i-th detection point andy_(t,i) is a second coordinate of the i-th detection point.

The range rate equation for each detection point can then be expressedas:{dot over (r)} _(i,cmp)=(y _(t,COR,) −y _(t,i))ω_(t) cos θ_(i)+(x _(t,i)−x _(t,COR))ω_(t) sin θ_(i),wherein this equation can be reduced to:{dot over (r)} _(i,cmp)=(y _(t,COR))ω_(t) cos θ_(i)+(−x _(t,COR))ω_(t)sin θ_(i),because ofy _(t,i) cos θ_(i) =r _(t,i) sin θ_(i) cos θ_(i) =x _(t,i) sin θ_(i).

Recall that the range rate equation is generally defined as:

${\overset{.}{r}}_{i,{cmp}} = {{\begin{bmatrix}{\cos\;\theta_{i}} & {\sin\;\theta_{i}}\end{bmatrix}\begin{bmatrix}c_{t} \\s_{t}\end{bmatrix}}.}$

A comparison with the formulation of the range rate equation whichincludes the yaw rate shows that the estimated first and secondcoefficients can be expressed as{tilde over (c)} _(t)=(y _(t,COR))ω_(t){tilde over (s)} _(t)=(−x _(t,COR))ω_(t),respectively. Therefore, the velocity of the i-th detection point can beexpressed as:V _(t,i) ^(x) ={tilde over (e)} _(t)+(−y _(t,i))ω_(t)V _(t,i) ^(y) ={tilde over (s)} _(t)+(x _(t,i))ω_(t).

The yaw rate is usually unknown but may be estimated. Taking intoaccount such an estimation, the estimated velocity at the i-th detectionpoint can be expressed as:

${{\hat{V}}_{i,t} = {\begin{bmatrix}{\hat{V}}_{t,i}^{x} \\{\hat{V}}_{t,i}^{y}\end{bmatrix} = {\begin{bmatrix}{\overset{\sim}{c}}_{t} \\{\overset{\sim}{s}}_{t}\end{bmatrix} + {\begin{bmatrix}{- y_{t,i}} \\x_{t,i}\end{bmatrix}\lbrack {\hat{\omega}}_{t} \rbrack}}}},$wherein the velocity portion with respect to the yaw rate can beidentified as:

${{\hat{V}}_{i,\omega} = {\begin{bmatrix}{- y_{t,i}} \\x_{t,i}\end{bmatrix}\lbrack {\hat{\omega}}_{t} \rbrack}},$with a second coordinate −y_(t,i) in the first spatial dimension x and afirst coordinate x_(t,i) in the second spatial dimension y, i.e. thefirst and second coordinates define the position of the i-th detectionpoint as (x_(t,i), y_(t,i)) with the second coordinate being inverted.

In a more compact notation the estimated velocity at the i-th detectionpoint can be expressed as:{circumflex over (V)} _(i,t) =

+{circumflex over (V)} _(i,ω),wherein this estimated velocity can be set equal to the estimatedcompensated range rate of the velocity profile equation, as addressedfurther above.

The uncertainty estimate of the estimated velocity for the i-thdetection point is preferably defined as:Û _(V) _(i,t) ² =Û _(VP) ² +Û _(V) _(i,ω) ²,wherein Û_(V) _(i,t) ² is a two-dimensional matrix, Û_(VP) ² is atwo-dimensional matrix and a first portion of the uncertainty estimateÛ_(V) _(i,t) ², and Û_(V) _(i,ω) ² is a two-dimensional matrix and asecond portion of the uncertainty estimate Û_(V) _(i,t) ². The estimateas such, as well as the both portions are preferably squared figures,which avoid negative values for the estimate. The matrices alsopreferably represent dispersions with respect to the first and secondspatial dimensions. Although the uncertainty estimate is defined here asa sum of the first and second portion other combinations of the firstand second portion are possible in order to determine the uncertaintyestimate.

The first portion Û_(VP) ² represents the uncertainty with respect tothe first estimated coefficient {tilde over (c)}_(t) and the secondestimated coefficient {tilde over (s)}_(t) of the velocity profileequation. Thus, the first portion can be interpreted to represent theuncertainty with respect to the velocity profile solution. This can beexpressed as:

${{\hat{U}}_{VP}^{2} = \begin{bmatrix}{\hat{U}}_{c_{t}}^{2} & {\hat{U}}_{c_{t}s_{t}}^{2} \\{\hat{U}}_{s_{t}c_{t}}^{2} & {\hat{U}}_{s_{t}}^{2}\end{bmatrix}},$wherein Û_(c) _(t) ² is the uncertainty estimate of the first estimatedcoefficient, Û_(s) _(t) ² is the uncertainty estimate of the secondestimated coefficient, and Û_(c) _(t) _(s) _(t) ²=Û_(s) _(t) _(c) _(t) ²the cross-uncertainty estimate between the first and second estimatedcoefficient. In this way, the first portion corresponds in general to acovariance matrix.

The first portion can be further defined as:Û _(VP) ²={circumflex over (σ)}_(VP) ²+

_(VP) ²,with a covariance portion {circumflex over (σ)}_(VP) ² representing acovariance matrix of the first estimated coefficient and the secondestimated coefficient, and a bias portion

_(VP) ² representing a bias of the first estimated coefficient and thesecond estimated coefficient.The covariance portion can be expressed as:

${{\hat{\sigma}}_{VP}^{2} = \begin{bmatrix}{\hat{\sigma}}_{c_{t}c_{t}} & {\hat{\sigma}}_{c_{t}s_{t}} \\{\hat{\sigma}}_{s_{t}c_{t}} & {\hat{\sigma}}_{s_{t}s_{t}}\end{bmatrix}},$wherein {circumflex over (σ)}_(c) _(t) _(c) _(t) is the varianceestimate of the first estimated coefficient, {circumflex over (σ)}_(s)_(t) _(s) _(t) is the variance estimate of the second estimatedcoefficient, and {circumflex over (σ)}_(c) _(t) _(s) _(t) ={circumflexover (σ)}_(s) _(t) _(c) _(t) is the covariance estimate of the first andsecond estimated coefficient.

As indicated before the covariance portion is preferable determined as:{circumflex over (σ)}_(VP) ²={circumflex over (σ)}_({dot over (r)}) ²(X^(T) X)⁻¹.

Having regard to the bias portion, a general definition can be given as:

_(VP) ² =f(X,Y,k),wherein Y represents the compensated range rate {dot over (r)}_(cmp) andk represents some constants. Moreover, the matrix X is the same asbefore, namely:

$X = {\begin{bmatrix}{\cos\;\theta_{1}} & {\sin\;\theta_{1}} \\\vdots & \vdots \\{\cos\;\theta_{n}} & {\sin\;\theta_{n}}\end{bmatrix}.}$In particular, the bias portion can be defined as:

_(VP) ² =k _(ols_bias_scale){circumflex over (σ)}_(VP) ² +B,with

${B = \begin{bmatrix}k_{c\;\_\;{va}\; r\;\_\;{bias}} & 0 \\0 & k_{s\;\_\;{va}\; r\;\_\;{bias}}\end{bmatrix}},$wherein k_(ols_bias_scale) is a scaling calibration parameter,k_(c_var_bias) is an offset calibration parameter for the firstestimated coefficient, and k_(s_var_bias) is an offset calibrationparameter for the second estimated coefficient.

It is noted that the bias portion

_(VP) ² is preferably a function of the covariance matrix {circumflexover (σ)}_(VP) ² of the first and second estimated coefficient andadditional calibration parameters.

Having regard to the second portion Û_(V) _(i,ω) ² of the uncertaintyestimate the second portion can be defined as:

${{\hat{U}}_{V_{i,\omega}}^{2} = {{\begin{bmatrix}{- y_{t,i}} \\x_{t,i}\end{bmatrix}\lbrack {\hat{U}}_{\omega}^{2} \rbrack}\begin{bmatrix}{- y_{t,i}} & x_{t,i}\end{bmatrix}}},$wherein Û_(ω) ² is the estimated uncertainty of the yaw rate.

In order to avoid a dynamic estimation of the uncertainty of the yawrate, it is possible to rely on a predetermined uncertainty. This can bedone under the assumption that yaw rate of objects is bounded. Forexample the yaw rate of typical traffic objects, e.g., vehicles usuallymay not exceed 30 degrees per second. It is then possible to model theyaw rate as a distribution (probability density function=pdf), forexample a uniform distribution with zero mean and a maximum valueω_(t_max) of the yaw rate ω_(t) as:

${{pdf}( \omega_{t} )} = \{ {\begin{matrix}\frac{1}{2\omega_{t\;\_\;{ma}\; x}} & {{{for}\mspace{14mu}\omega_{t}} \in \lbrack {{- \omega_{t\;\_\;{ma}\; x}},\omega_{t\;\_\; m\;{ax}}} \rbrack} \\0 & {otherwise}\end{matrix}.} $The maximum value of the yaw rate (also called extremum) can bepredetermined by a calibration parameter as:ω_(t_max) =k _(max_yaw_rate).From standard mathematics the variance of the uniform pdf is:

$\sigma_{\omega_{t}}^{2} = {\frac{\omega_{t\;\_\;{ma}\; x}^{2}}{3}.}$The uncertainty of the yaw rate can then be set to the variance, thisis:U _(ω) _(t) ²=σ_(ω) _(t) ².Therefore, the second portion of the uncertainty estimate ispredetermined and expressed as:

${U_{V_{i,\omega}}^{2} = {{\begin{bmatrix}{- y_{t,i}} \\x_{t,i}\end{bmatrix}\lbrack U_{\omega_{t}}^{2} \rbrack}\begin{bmatrix}{- y_{t,i}} & x_{t,i}\end{bmatrix}}},$wherein it is understood that the second portion represents theuncertainty of the estimated velocity with respect to an angularvelocity, the first coordinate x_(t,i) in the second spatial dimension yand the second coordinate y_(t,i) in the first spatial dimension x.

FIG. 6 gives an overview of some aspects of the method described above.Each of the boxes corresponds to an exemplary step of the method,wherein the step is indicated inside the box. Dashed boxes indicate thatthe corresponding step is merely optional.

The proposed uncertainty estimate has been evaluated in two differentexample scenarios. In order to quantify the validity of the uncertaintyestimate the Normalized Estimation Error Squared (NEES) is used as ametric. This metric can be generally interpreted as a measure ofconsistency between the estimated velocity and an estimated variance oruncertainty estimate. A general definition can be given as:e _(i)=({dot over (r)} _(i,cmp) −{dot over ({circumflex over (r)})}_(i,cmp))^(T) {circumflex over (P)} _(i) ⁻¹({dot over (r)} _(i,cmp)−{dot over ({circumflex over (r)})} _(i,cmp)),with {circumflex over (P)}_(i) ⁻¹ representing the inverse of anestimated covariance matrix and e_(i) representing the NEES for the i-thdetection point. The covariance matrix {circumflex over (P)}_(i) iseither the estimated covariance matrix {circumflex over (σ)}_(VP) ² orthe proposed uncertainty estimate Û_(V) _(i,t) ².

The estimated velocity and the estimated covariance matrix areconsistent if the expected value of NEES is equal to the dimension n ofthe covariance matrix (here n=2):H ₀ :E(e _(i))=n.

In both example scenarios a simulation of a moving object has beencarried out with 1000 Monte Carlo iterations. In the first scenario, astraight moving object was simulated. When using the estimatedcovariance matrix {circumflex over (σ)}_(VP) ² the expected NEES isE(e_(i))=3.01 which is inconsistent at the 95% significance level. Whenusing the proposed uncertainty estimate E(e_(i))=2.03 which isconsistent at the 95% significance level.

In the second scenario a yawing object was simulated. When using theestimated covariance matrix {circumflex over (σ)}_(VP) ² E(e_(i))=1663.4which is completely inconsistent. Using this as an “uncertaintyestimate” would be dangerous for a tracking application. However, whenusing the proposed uncertainty estimate then E(e_(i))=2.04 which is wellconsistent at the 95% significance level. Therefore, the proposeduncertainty estimate can be used, e.g., for safe tracking applications.

LIST OF REFERENCE SIGNS

-   1 origin of world coordinate system-   2 target vehicle-   3 front bumper-   3′ rear bumper-   3″ origin of vehicle coordinate system-   4 host vehicle-   5 origin of sensor coordinate system-   5′ radar system-   6, 6′ detection point-   7 center of rotation of the target

We claim:
 1. A method comprising: determining an uncertainty estimate ofan estimated velocity of an object, the uncertainty estimaterepresenting an uncertainty of a dispersion of the estimated velocitywith respect to a true velocity of the object, wherein determining theuncertainty estimate comprises: determining a first portion of theuncertainty estimate, the first portion representing the uncertaintywith respect to a dispersion of a first estimated coefficient and asecond estimated coefficient of a velocity profile equation of theobject, the first estimated coefficient being assigned to a firstspatial dimension of the estimated velocity and the second estimatedcoefficient being assigned to a second spatial dimension of theestimated velocity, the velocity profile equation representing theestimated velocity in dependence of the first estimated coefficient andthe second estimated coefficient; determining a second portion of theuncertainty estimate based on a predetermined intermediate; secondportion representing the uncertainty with respect only to a dispersionof an angular velocity of the object, a first coordinate of the objectin the second spatial dimension, and a second coordinate of the objectin the first spatial dimension; and determining the uncertainty estimatebased on the first portion and the second portion.
 2. The methodaccording to claim 1, wherein: the uncertainty estimate and one or moreof the first portion and the second portion of the uncertainty estimateare determined as a two-dimensional matrix; and the two-dimensionalmatrix represents a dispersion with respect to the first spatialdimension and the second spatial dimension.
 3. The method according toclaim 1, wherein the first portion is determined based on: a covarianceportion representing a covariance matrix of the first estimatedcoefficient and the second estimated coefficient; and one or more of abias portion representing a bias of the first estimated coefficient andthe second estimated coefficient.
 4. The method according to claim 3,wherein: the bias portion is determined based on a plurality ofdetection points of the object and at least one constant; and each ofthe plurality of detection points comprises an estimated velocityassigned to a detection position on the object, the detection positionon the object being represented at least by an angle.
 5. The methodaccording to claim 1, wherein: the estimated velocity is assigned to aposition of the object represented by the first coordinate of the objectin the first spatial dimension and the second coordinate of the objectin the second spatial dimension; and the uncertainty estimate isdetermined in dependence of the position of the object.
 6. The methodaccording to claim 1, wherein the intermediate second portion ispredetermined by a variance of a distribution of the angular velocity ofthe object.
 7. The method according to claim 6, wherein the distributionis a uniform distribution with at least one predetermined extremum ofthe angular velocity of the object.
 8. The method according to claim 1,wherein the intermediate second portion is predetermined by at least oneparameter representing an extremum of the angular velocity of theobject.
 9. The method according to claim 1, wherein the uncertaintyestimate is determined based on a sum of the first portion and thesecond portion.
 10. The method according to claim 1, further comprising:controlling a vehicle in a vicinity of the object in dependence of theestimated velocity of the object, wherein the estimated velocity isprocessed in dependence of the uncertainty estimate.
 11. The methodaccording to claim 1, further comprising: tracking the object based onthe uncertainty estimate; classifying the object based on theuncertainty estimate; and discriminating the object from other objectsbased on the uncertainty estimate.
 12. A system comprising: at least oneprocessor configured to determine an uncertainty estimate of anestimated velocity of an object, the uncertainty estimate representingan uncertainty of a dispersion of the estimated velocity with respect toa true velocity of the object, wherein the processor is configured todetermine the uncertainty estimate by: determining a first portion ofthe uncertainty estimate, the first portion representing the uncertaintywith respect to a dispersion of a first estimated coefficient and asecond estimated coefficient of a velocity profile equation of theobject, the first estimated coefficient being assigned to a firstspatial dimension of the estimated velocity and the second estimatedcoefficient being assigned to a second spatial dimension of theestimated velocity, the velocity profile equation representing theestimated velocity in dependence of the first estimated coefficient andthe second estimated coefficient; determining a second portion of theuncertainty estimate based on a predetermined intermediate secondportion representing the uncertainty with respect only to a dispersionof an angular velocity of the object, a first coordinate of the objectin the second spatial dimension, and a second coordinate of the objectin the first spatial dimension; and determining the uncertainty estimatebased on the first portion and the second portion.
 13. The systemaccording to claim 12, wherein: the processor is configured to determinethe uncertainty estimate and one or more of the first portion and thesecond portion of the uncertainty estimate as a two-dimensional matrix;and the two-dimensional matrix represents a dispersion with respect tothe first spatial dimension and the second spatial dimension.
 14. Thesystem according to claim 12, wherein the processor is configured todetermine the first portion based on: a covariance portion representinga covariance matrix of the first estimated coefficient and the secondestimated coefficient; and one or more of a bias portion representing abias of the first estimated coefficient and the second estimatedcoefficient.
 15. The system according to claim 14, wherein: theprocessor is configured to determine the bias portion based on aplurality of detection points of the object and at least one constant;and each of the plurality of detection points comprises an estimatedvelocity assigned to a detection position on the object, the detectionposition on the object being represented at least by an angle.
 16. Thesystem according to claim 12, wherein the processor is configured to:assign the estimated velocity to a position of the object represented bythe first coordinate of the object in the first spatial dimension andthe second coordinate of the object in the second spatial dimension; anddetermine the uncertainty estimate in dependence of the position of theobject.
 17. The system according to claim 12, wherein the processor isconfigured to: predetermine the intermediate second portion by avariance of a distribution of the angular velocity of the object. 18.The system according to claim 17, wherein the distribution is a uniformdistribution with at least one predetermined extremum of the angularvelocity of the object.
 19. The system according to claim 12, whereinthe processor is configured to: predetermine the intermediate secondportion by at least one parameter representing an extremum of theangular velocity of the object.
 20. The system according to claim 12,wherein the processor is configured to: determine the uncertaintyestimate based on a sum of the first portion and the second portion. 21.The system according to claim 12, wherein the processor is furtherconfigured to: control a vehicle in a vicinity of the object independence of the estimated velocity of the object, wherein theestimated velocity is processed in dependence of the uncertaintyestimate.
 22. The system according to claim 12, wherein the processor isfurther configured to: track the object based on the uncertaintyestimate; classify the object based on the uncertainty estimate; anddiscriminate the object from other objects based on the uncertaintyestimate.